## Anecdote

It is Boxing Day, the morning after the National Football League’s (NFL) marquee matchup of its Week 16 schedule (and perhaps the entire regular season). The Baltimore Ravens have just pulled off a surprising 33-19 upset of the juggernaut San Francisco 49ers. Leading up to the game, the contingent of highly skilled bookmakers (experts tasked with setting the odds for the match) had forecasted that the most likely outcome would be the 49ers winning the match by 6 points. In other words, the Ravens were 6-point underdogs.

Two professional bettors – Jack Mansion and Rasheed Parker – are discussing the match. Mansion and Parker both placed wagers on the match, with Parker winning his Ravens (+6) wager and Mansion losing his 49ers (-6) bet. Parker’s judgment has been rewarded with a profit of 91% on his stake. Mansion’s entire wager is lost (a loss of 100%). This is the essence of point spread betting, the most popular type of wager in North American sports. It is a notoriously difficult market to beat, despite the fact that doing so “only” requires an accuracy of 52.4%.

Having won his wager, Parker exclaims:

*“Great game last night. I told you that you should’ve taken the Ravens!”*

Feeling slightly agitated and discouraged, Mansion responds:

*“Brock Purdy just didn’t have it last night, and he was quite unlucky with some of those interceptions. I still feel like the 49ers were the correct side of the wager.”*

**Who is correct in this exchange? **

The somewhat frustrating reality is that we will never know.

## Where does randomness come from?

There are numerous sources of randomness that collectively exert their influence on in-game outcomes and, ultimately, the final score.

Every time that a ball is thrown by the quarterback (or kicked by a punter or place kicker), the turbulent forces of wind and air resistance alter the path of the ball. In turn, this influences whether the thrown ball makes its way into the receiver’s hands (in the event of a pass), or crosses through the uprights (during kicks). Even if one knows the wind forecast, the forces of wind cannot be predicted beforehand: at the level of individual plays, there is no available information that can be used to forecast the outcome.

While the example of weather is an obvious source of randomness, there are numerous others. Every time that the pigskin slips out of a player’s hands and hits the playing surface (a “fumble”), the direction that the ball takes as it bounces off the field is unpredictable. Whether the ball is recovered by an offensive or defensive player has a meaningful impact on the match’s outcome. As my probability theory professor noted on the first day of class, although the laws of physics may predict whether a tossed coin will land heads or tails if one follows the trajectory of the coin as it descends towards the ground, it is far more useful to simply conclude that the two outcomes are equally likely. Although the identity of the team that recovers the fumble may not be equally likely, a similar argument holds here: there is some probability *p* that the team that fumbled the ball recovers it, and another probability *1-p* that a player on the opposing team lands on it first.

Some sources of randomness manifest before the match even begins. The quality of the players’ sleep may be disrupted by random events. As every parent knows, children have a way of surprising you with sleepless nights, sometimes even (systematically?) coinciding with important events in the parents’ lives occurring the following morning. On a somewhat related note, there is also the urban legend of opposing players procuring the services of attractive females to disrupt the sleep of their opponents in their hotel rooms on the eve of important matches. And just as I write this, there emerges a __report__ of a fire alarm going off in the hotel room of the San Francisco 49ers (who have subsequently played their way into SuperBowl 58 where they will shortly face off against the defending champion Kansas City Chiefs) in the early hours of Thursday, just 3 days shy of the day of the big game.

## What are the consequences of randomness?

There is an essentially infinite number of random events that manifest either before or during the match. It is not difficult to see that these various sources of randomness are *independent*: the outcome of the wind pushing Brock Purdy’s pass has no influence on the direction that a fumble takes later in the game. These different outcomes collectively shape the game script and determine the associated margin of victory (as well as the total number of points scored, and really any outcome that one likes to wager on).

The precise mathematical relationship between the various random phenomena and the final margin of victory is complex and virtually impossible to model. What we *can* say is that the set of potential final scores ranges from approximately -50 (visiting team wins by 50 points) to +50 (home team wins by 50). Unlike the toss of a coin or the roll of a dice, however, the different outcomes are not equally likely. Certain margins of victory are far more likely to occur than others, for example -3, +3 and +7.

The quality of the two competing teams certainly bears an influence on which outcomes are more likely to occur than others. But this most obvious variable – how much better (or worse) the home team is relative to the visiting team – is then “superimposed” with the multitude of random forces described above.

What I would like to convey here is that, because of the collective activity of these random events, the margin of victory in a football match is not deterministic. *Instead, the final score should be thought of as being drawn from a distribution.*

## An example

Over the last twenty years, there have been 185 regular season NFL matches with the home team favored by exactly 6 points (according to bettingdata.com’s “consensus” point spread values). It is instructive to look back and note what the actual outcomes of those matches were: on average, how close were the final margins of victory to the bookmakers’ estimate (i.e., the point spread, which is 6 for all of these matches)?

Figure 1 below shows the distribution (a histogram) of these 185 outcomes. Each bar corresponds to one of the potential outcomes (i.e., an integer ranging from -30 to +52). The height of each bar represents how often that specific margin actually occurred.

Perhaps surprisingly, the most common outcome from this subset of matches was the home team winning by 3 points, which occurred a total of 15 times (approximately 8% of the time). The next most frequent was the *visiting team* (the underdog) winning by 6 points (12 occurrences). There were only 6 matches where the actual margin of victory matched the point spread exactly – in these games, all bets were returned, referred to as a “push” in betting parlance. In terms of extremes, the most lopsided upset occurred in Week 17 of the 2015 season (Seahawks defeated the Cardinals 36-6), and the most one-sided cover was the Rams 52-0 shellacking of the Raiders in Week 13 of 2014.

It is clear from the extent of the histogram that there is a very large amount of variability that enters into the margin of victory of NFL games. The average absolute deviation between the actual outcome and the sportsbook point spread was 10.9 points. There was a slight bias towards the underdogs winning the matches *against the spread* (103 of 185 or 56%). This is also reflected in the median margin of victory of 4 points, a deviation of 2 from the book estimate.

*Figure 1. The distribution of margin of victory for the last 185 NFL matches in which the home team was favored to win by exactly 6 points. The burgundy bar corresponds to the December 26 contest between the visiting Baltimore Ravens and San Francisco 49ers (Ravens won by 14 points). The most frequent outcome was the home team winning by 3 points (15 instances).*

The margin in the Boxing Day clash between the 49ers and Ravens (-14) occurs in the left “tail” of the distribution (burgundy shaded bar), potentially signifying that this particular outcome was an outlier (a low probability event). However, note that this sort of argument relies on the assumption that the December 26 outcome was in fact drawn from *this* particular distribution.

## We only observe one outcome

In reality, we only ever observe one *realization* of the match. It is useful to think of the game script and resulting margin of victory as unfolding analogously to the roll of a dice. Instead of the six possible outcomes (all of which are equally likely), however, this hypothetical “football dice” turns up outcomes in proportion to their likelihood. In the example above, an outcome of +7 is far more likely to be observed then, say, +42.

It is critical to understand that each “football dice” is unique, reflecting the circumstances and context defining each match:

The identity of the two teams

The date and time of the match

The weather at gametime

The injury report

And many more

What this means is that the December 26 Ravens - 49ers match will never be replicated, and the alternative outcomes will not be witnessed.

## Forecasting the distribution

If I magically gave you the distribution of potential outcomes leading up to the Ravens - 49ers match, how would you use it?

You would probably first look to see where the point spread is on the distribution. If it is located to the left of the distribution’s “middle” (the median, not the same as the mean!), you would likely conclude that betting on the home team is the correct wager. On the other hand, if the spread is located in the right tail of the distribution, you would likely elect to place a wager on the visiting team. Indeed, if you were forced to wager on the match, the wiser decision is to bet on the side which has more “mass” under the probability distribution – here, “side” refers to a partition of the distribution by the point spread, with the right side corresponding to the home team winning against the spread and the left side to the visiting team.

However, what if the spread happens to coincide exactly with the distribution’s median? Or what if it is very close to it? In this scenario – not an uncommon one – the optimal decision is, in the parlance of sports wagering, to “stay away” and avoid wagering altogether.

The reason is that, regardless of the outcome, you will lose money to the sportsbooks in the long run if you continue wagering on such games. The intuition here is that the proportion of bets that you win will not be enough to compensate for the sportsbook’s commission, *even if you always wager on the side with the higher probability of winning the match*!

The thought experiment above is all but hypothetical. One will never be armed with knowledge of the game’s underlying probability distribution, although sportsbooks and astute bettors spend much of their time *estimating* just that. One may reasonably pose the question of how one can estimate the distribution of an event that is only witnessed one time? This is a deep and complex problem that I will address in a future post. For now, it suffices to say that by combining the abundance of historical data, knowledge of the dynamics of football and the composition of the two teams, it is possible to generate forecasts of the distribution of margin of victory for an upcoming match. One can then use the distribution to guide one’s betting decision.

Which naturally begs the question: how will we know whether it was the correct one?

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